Andy, I'm so glad to hear that you and your group are considering going to Switzerland July 6, as opposed to June 15 -- It makes a huge difference! In July all the hiking trails have gotten fresh maintenance and opened after being covered in melting snow piles into June, and all the cable cars are running again. The grass in the pastures has all greened-up and the colorful alpine flowers are in full bloom. The temperatures are mild -- no more cold waves like in June.
The trick to getting the best views in July is plan to start your outings as soon as the cable cars open in the morning. That's usually when you get the clearest sky that you'll get all day. I like to use the weather app MeteoSwiss on my iPhone to plan my day, but keep in mind that weather out there is so complicated that it isn't reliably predictable until the night before or the morning of. Here's a link to that website:
http://www.meteoswiss.admin.ch/home.html?tab=overview
My other best-views trick is to schedule at least twice as many days there as I feel I need, for those periods when clouds roll in and completely blanket everything, to where you can't see the mountains. But partly cloudy days are just fine! It's so stunning and beautiful that it's worth any wait.
My favorite hiking trails aren't open or at their best until around the beginning of July. Among my easy favorites are Männlichen-to-Kleine Scheidegg (take the cable car from Wengen up to Männlichen), and Kleine Scheidegg-to-Wengernalp (take the cog-rail train from Wengen up to Kleine Scheidegg). (I don't recommend taking the gondola from Grindelwald/Grund up to Männlichen because when I took it this past summer (July 2017) the windows were so scratched up that I could barely see the view.) Both of these hikes go through vivid green pastures full of flowers with views in your face of snow-capped mountains (Jungfrau, Eiger, Monch, Silberhorn). Near Wengernalp are places that sell cheese made from the cows you've passed. Btw, my favorite outdoor cafe in Kleine Scheidegg is the Grindelwaldblick (also recommended in the Rick Steves Switzerland book.
My other favorite trail is the Northface Trail, (a medium difficulty hike) which starts from Allmendhubel -- take the funicular from Mürren up to Allmendhubel and follow the blue marked signs through spectacular green mountains with views of a range of snow-capped mountains, like the Jungfrau, eventually making your way to Mürren.
Detailed instructions are in the Rick Steves Switzerland book.
When the weather doesn't allow good views, you can see loads of waterfalls that are pouring down from the cliffs, by walking through the Lauterbrunnen Valley from the village of Lauterbrunnen. Or, if it's raining, visit Trummelbach Falls, which is a short bus ride between Lauterbrunnen and Stechelberg, in the Lauterbrunnen Valley. That series of waterfalls is inside a mountain! Another no-view-period special experience is to take a short ferryboat ride on Lake Brienz to visit the very beautiful Giessbach Falls. To get there, take the train from Lauterbrunnen to Interlaken Ost, change to take the train to the town of Brienz nearby. At Brienz get off the train and the lake ferry will be docked there beside the train station. Take the ferry to its first stop, which is for Giessbach Falls. Take the funicular up the cliff to Grand Hotel Giessbach and walk the path along the waterfalls.
Another no-view suggestion is to take a train to Luzern for the day.
You'd asked about the difference between the trail from Allmendhubel to Grutchalp, vs. the trail from Mürren to Grutchalp. The path from Mürren to Grutchalp is flat and follows the level train track. In contrast, the Allmendhubel trail starts up high and involves a descent, and the view is of the mountains from higher up. I didn't walk all the way down to Grutchalp, but instead descended more quickly down to Winteregg where I caught the train to Grutchalp (after buying some local cheese.)
